Before you head to Settings> General> Software Update to download iOS 6, it's a good move to first back up your iPhone to iCloud. Doing so will make it much, much easier to recover from any problems, if any crop up during the install. Performing an iCloud backup saves all of the data on your device, including details like app placement and settings, to Apple's online servers. (You can also plug your device into a computer and back up using iTunes. Be sure to download iTunes 10.7 first.) Backing up is not a step I would skip.
Once that's done, you can begin the update. If the battery is at less than 50% capacity, you will have to plug your device in or the install won't begin.
You can also upgrade from a computer running iTunes, and here there are two options: Upgrade (which leaves existing data in place) or Restore (which erases content and brings the device back to default settings).
After iOS 6 is installed and the device has restarted, you'll see a welcome screen and "Welcome" cycling through different languages in the slide-to-unlock area. Once you unlock, you go to a Setup Assistant similar to that used in OS X. After choosing your language and country/region, connecting to a Wi-Fi or cellular network and activating the phone, you're asked to decide whether you want Location Services on or off and sign into iCloud.
If you chose to restore the iPhone and start from scratch, this is where you'll be thankful for the aforementioned backup: Just sign in and choose Restore from iCloud or iTunes.That brings back all of your data and settings as if nothing happened, right down to app placement. By the way, if you're restoring from an iCloud backup, you can initiate one over cellular data, such as LTE, but I highly recommend Wi-Fi; otherwise, you may use up your monthly data allowance in one shot.
As apps begin to download, if you need to use a specific app immediately, tap its icon to bring that app to the front of the download queue and it'll begin transferring in moments.
I should note a minor glitch I came across, in case you encounter it: At one point, I was given a warning that the Restore was incomplete, and that I should plug the phone back to iTunes to finish. After tapping OK, the iPhone continued to restore itself from my iCloud backup. I didn't have to connect to iTunes to install any apps.